linear time invariant systemLinearTimeInvariantLTISystems2004-05-20 23:41:502006-09-16 21:54:24DefinitionLTI% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA \emph{linear time invariant system} (LTI) is a linear dynamical system $T(p)$,
\begin{align*}
y(k) &= T(p) \; u(k),
\end{align*}
with parameter $p$ that is time independent. $y(k)$ denotes the
system output and $u(k)$ denotes the input. The independent variable
$k$ can be denoted as time, index for a discrete sequences or
differential operaters (e.g. such as $s$ in Laplace domain or $\omega$
in frequency domain).
For example, for a simple mass-spring-dashpot system, the system
parameter $p$ can be selected as the mass $m$, spring constant $k$ and
damping coefficient $d$. The input $u$ to the said system can be chosen
as the force applied to the mass and the output $y$ can be chosen as the
mass's displacement.
LTI system has the following properties.
\begin{description}
\item[Linearity:]
If $y_1 = T x_1$ and $y_2 = T x_2$, then
$$T \{\alpha x_1 + \beta x_2 \} = \alpha y_1 + \beta y_2 $$
\item[Time Invariance:]
If $y(k) = T x(k)$, then
$$ y(k+\delta_k) = T x(k + \delta_k) $$
\item[Associative:]
$$ T_1 \cdot ( T_2 \cdot T_3 ) = (T_1 \cdot T_2) \cdot T_3 $$
\item[Commutative:]
$$ T_1 \cdot T_2 = T_2 \cdot T_1 $$
\end{description}
A LTI system can be represented with the following:
\begin{itemize}
\item Transfer function of Laplace transform variable $s$, which is commonly
used in control systems design.
\item Transfer function of Fourier transform variable $\omega$, which is
commonly used in communication theory and signal processing.
\item Transfer function of z-transform variable $z^{-1}$, which is
commonly used in digital signal processing (DSP).
\item State-space equations, which is commonly used in modern control
theory and mechanical systems.
\end{itemize}
Note that all transfer functions are LTI systems, but not all
state-space equations are LTI systems.